A Concept of Energy Banach Space and Its Application to the Gradient Method1

The notion of energy Banach space is introduced for nonlinear operators and some properties are discussed. The concept is used to extend the gradient method for operators which are not continuous themselves but their transform to a suitable energy space has appropriate continuity and monotonicity properties. Convergence is obtained in the energy norm. The obtained method is applied to quasilinear elliptic boundary value problems